Blender: Project a map on a sphere
Considering the map part corresponds to an equirectangular projection, we can do a plane to sphere deformation by the approach indicated here.
Making a sphere from a grid
The base grid should be in the proportions of width/height = 2 as we map as longitude is 360 degrees and longitude is 180 degrees.
If the grid is parallel to the front view, we can give it some flat subdivisions and bend it 180° around X and 360° around Z.
To unwrap it (as we'll use UV mapping), still in front view and edit mode use U then &quot;Project from view (bounds)&quot;.

The shader to place the map part
From the given image, we can see that the map part is (approximatively) from -160° to -126° in longitude and from 28° to 64° in latitude.
What we have to do:

Determinate if a given point is inside the map part
Remap this map part at the position given by its coordinates ranges

In X (or U) in the UV map, a given coordinate is between 0 and 1. The longitude is between -180 and 180. So we can shift the map part coordinate by 180 and divide it by 360 to go from the longitude space to the UV space.
Same thing from Y (or V).
With this calculation (done by AddDivide node group in the blend below) applyed to the map part coordinates, we can test if a given UV map point is inside the map (done by Between node group in the file below).
This first calculation indicates if the texture has to be drawn on the sphere for a given UV map point.
Now to draw the texture at the wanted location, we need to remap the UV point to the texture space in the good location and proportion.
This is done by substracting the min map coordinate in UV space with UV point coordinate then divide that by the map range in UV space.
X in texture space = (X in UV space - min map X in UV space) / (max map X - min map X in UV space)

This part is done by Remap01 node group in the file.
Last, we combine all together in a single node group so that you can apply it to several map parts.

This node group is called RemapEquirectangular and has the following parameters: the UV coordinates, min and max longitude, min and max latitude.
The result is the following: